The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

This discussion is now closed.

Check out other Related discussions

Simple definitions, need some help explaining terms! Rep for help

I'll give some rep if you can come up with some nice explanations (definitions of the following terms):

Eigenfunction
Eigenvector
Basis set
Orthonormal basis set
Complete basis set
Operator

Cheers guys!
Reply 1
Avatar for Notnek
Notnek
21
I do maths so I dont know if my definitions will be perfect for you.

Basis set: A linear independent spanning set of a given vector space.

Orthonormal basis set: A basis set where each element is orthogonal to every other element

Eigenvector: A vector is an eigenvector x for a square matrix A if there exists a non-zero number lambda such that: Ax = lambda(x)

I'm not confident about the other terms so I'll let someone else define them.

EDIT: Are you looking for explanations or definitions. I've provided mathematical definitions which may not explain the terms.
Reply 2
Avatar for Kolya
Kolya
17
I dislike the order that the definitions are given in, so I will rearrange them.

A set of vectors is a basis set of a vector space V over a field K if the vectors are linearly independent - i.e. none of the vectors can be made as a linear combination of another, using scalars in K - and they span V (i.e. every v∈Vv \in V is a linear combination of vectors in the basis set).

A basis is orthonormal if each basis vector has unit norm and are all orthogonal to each other.

A linear operator, T:U→VT: U \to V associates each vector in U with some vector in V, such that T(αu1+βu2)=αT(u1)+βT(u2)T(\alpha u_1 + \beta u_2) = \alpha T(u_1) + \beta T(u_2) (where u1,u2∈Uu_1, u_2 \in U and α,β\alpha , \beta are scalars from the field over which the vector space exists.)

An eigenvector of a linear operator is a vector that, when transformed by that operator, becomes a multiple of itself. (in Maths language, a vector x is an eigenvector iff Ax=λxAx = \lambda x for some scalar lambda.)

An eigenfunction is a function f(x)f(x) is similar to an eigenvector - it is a function that, when transformed by the linear differential operator, becomes a multiple of itself. (In Maths-language: a function that when acted upon by a linear differential operator L (i.e. L[f(x)]:=αn(x)f(n)+...+α1(x)f′+α0(x)f)L[f(x)] := \alpha _n (x) f^{(n)} + ... + \alpha _1 (x) f' + \alpha_0 (x) f)) such that L[f(x)]=λf(x)L[f(x)] = \lambda f(x) for some scalar lambda).

I'm not sure if this will help as it is difficult to explain every single concept in detail without collapsing, but, having read the definitions, are there any that you still do not understand?

(I think I might collapse anyway...)
Reply 3
Avatar for Kolya
Kolya
17
notnek
Orthonormal basis set: A basis set where each element is orthogonal to every other element
Don't they also need to be unit norm? It's an orthonormal basis set, not orthogonal.
Reply 4
Avatar for Notnek
Notnek
21
Kolya
Don't they also need to be unit norm? It's an orthonormal basis set, not orthogonal.

Yes of course - forgot about that.

Woooo 2000th post!

Related discussions

Latest

Trending

Trending

Articles for you